How do you find the local extrema for y = [1 / x] - [1 / (x - 1)]?

1 Answer
Mar 8, 2017

y has a local minimum of 4 at x=1/2

Explanation:

y=1/x - 1/(x-1)

= x^(-1) - (x-1)^(-1)

y' = -x^-2 + (x-1)^-2 *1 {Power rule and Chain rule]

y will have extrema where: y' =0

I.e. where: -x^-2 + (x-1)^-2 =0

(-(x-1)^2+x^2)/(x^2(x-1)^2)=0

-x^2+2x-1 + x^2 =0

x=1/2

As can be seen from the graph of y below, x=1/2 is local minimum.

graph{1/x - 1/(x-1) [-10.66, 11.84, -3.845, 7.405]}

Thus: y_min = y(1/2) = 1/(1/2) - 1/(1/2 -1)

= 2+2 =4